Variance-minimizing exponential retention
= Variance-minimizing exponential retention
{title2=$M^*=\mu$}
For <exponential distribution> claims of mean $\mu$ and <compound Poisson distribution> count parameter $\lambda$, the total party <variance> under <excess of loss reinsurance> is $g(M)=2\lambda\mu^2(1-(M/\mu)e^{-M/\mu})$. Its derivative is $2\lambda e^{-M/\mu}(M-\mu)$, so $M=\mu$ is the unique global minimum, with value $2\lambda\mu^2(1-e^{-1})$.